A wheel starting from rest is rotating with a constant angular acceleration of 2 rad / `sec^(2)` Interval .A student notes that it traces an angle of 80^ radian in 4 sec.interval. What was the angular velocity of the wheel , when the student started his observations ?

Options

A. `omega_(0) = 8 rad //s`
B. `omega_(0) = 16 rad //s`
C. `omega_(0) = 24 rad //s`
D. `omega_(0) = 48 rad //s`

Explanation

To solve the problem step by step, we can use the equations of rotational motion. Here’s how we can find the initial angular velocity of the wheel:

### Given Data:
- Angular acceleration, \( \alpha = 2 \, \text{rad/s}^2 \)
- Angular displacement, \( \theta = 80 \, \text{rad} \)
- Time, \( t = 4 \, \text{s} \)

### Step 1: Use the equation of angular displacement
We will use the equation of motion for rotational systems:
\[ \theta = \omega t + \frac{1}{2} \alpha t^2 \]
where:
- \( \theta \) is the angular displacement,
- \( \omega \) is the initial angular velocity,
- \( t \) is the time,
- \( \alpha \) is the angular acceleration.

### Step 2: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 80 = \omega (4) + \frac{1}{2} (2)(4^2) \]

### Step 3: Calculate \( \frac{1}{2} \alpha t^2 \)
Calculate \( \frac{1}{2} (2)(4^2) \):
\[ \frac{1}{2} (2)(16) = 16 \]

### Step 4: Substitute back into the equation
Now substitute this value back into the equation:
\[ 80 = 4\omega + 16 \]

### Step 5: Rearrange the equation to solve for \( \omega \)
Rearranging gives:
\[ 80 - 16 = 4\omega \]
\[ 64 = 4\omega \]

### Step 6: Solve for \( \omega \)
Now, divide both sides by 4:
\[ \omega = \frac{64}{4} = 16 \, \text{rad/s} \]

### Final Answer
The initial angular velocity of the wheel when the student started his observations was:
\[ \omega = 16 \, \text{rad/s} \]

---